THINK: Go back to the star-like image and manipulate it
to get a very high r-ratio. What does the figure (image)
look like? Try to get the ratio as high as you possibly can
without crossing the lines. (Java File)
All squares have r = 16. | Java File
All circles have r = 4*pi = 12.56... | Java File
All 4x5 (8x10) rectangles have r = 16. 2. | Java File
THINK:
Triangles are figures that are unique in this ratio and are relatively
easy to explore. The sketch
on the next page offers the ability to
explore a triangle between two parallel lines. Carefully note the area
of the triangle as you move the top vertex, what happens to the
perimeter? What about the ratio (r)?
After exploring general
triangles we can explore "special" triangles:
All 3-4-5 (6-8-10) rt. triangles have r = 24. | Java File
All equilateral triangles have r = 12*sqrt(3) = 20.78 | Java File
All 45-45-90 rt triangles have r = 12 + 8sqrt(2) = 23.3. | Java File
| JavaSketchpad File |
Conclusion: We have explored
the cancer ratio (r) with figures up to four sides. Establish the ratio
for sides with more than four and try to create a figure with a large
ratio and small ratio. What types of figures produce the largest
ratio? What types of figures produce the smallest ratios?
Think: Try
various ratios out with this figure to
experiment with the topics you just learned?